How to cast the shadow of the texture's alpha channel and not the actual geometry of the plane?
Blender internal can be a bit confusing in that regard. The objects casting the shadows do not determine how those shadows are received in other objects. The receiving objects determine the quality of the shadow.

By default the receiving objects show the shadow of the pane, not of the texture.



The receiving objects  (in your case the ground plane) have to be set to receive transparent.